Hi Dan,

> If I've been working too intensely on micro-music-math type problems

> or scores I can get a very nasty "white line fever" type reaction when

> I finally haul myself somewhere and try to and sleep.

> Sometimes these are acutely unpleasant -- kind of like a vicious

> dizzying loop. But I swear on occasion I've seen things a thousand

> times more complicated that I can cogitate expanding and contracting

> right before me just beyond the threshold of retention.

> Like I wasn't exhausted enough already! That's kind of like talking in

> your sleep and saying "I'm so tired" -- madness!

Yes, I can't remember what particularly I was working on at the time,

but I know one of the things I was trying to understand was

Godel's proof of his incompleteness theorem for arithmetic.

That kind of thing can easily set one into a kind of mind boggling

infinite recursion. Actually you can even get a bit dizzy if one

focusses really hard on it.

Other times it is geometrical things that attract ones attention

in this same compelling way, e.g. trying to figure out all the faces

of some complex geometrical figure and how they fit together

- not that well advised a thing to do while cycling!

Other times, can be abstract algebraic things, or a particular

step in a proof that one is trying to figure out, or a key concept

one needs to define in order to carry through a proof, and one

kind of keeps worrying at it like a dog with a bone, trying

to figure it out. Then one might suddenly see it, and it may

be something really easy actually, at that point.

I well remember getting involved in all of those at one time or

another.

Yes, and sleep is what often gives it a chance to sort itself

out, it is fairly common I think for a mathematician to go to bed

with a proof in a very incomplete state, and wake up with the

key idea, or a few key ideas, so that it is all finished bar

the writing out and polishing up of the presentation.

I'm not sure whether that happens in ones sleep, or just very fast

in the relaxed moments while one awakens. At any rate, something

goes on there that seems to be sometimes way beyond ones mormal

problem solving ability - in terms of the speed with which it happens

anyway.

I'm inclined to think it actually happens very fast at the moment

one wakes up.

Can also happen in other situations. A famous mathematician

at Oxford got the key idea that was the basis for one of the

theorems he is famous for while crossing a street with someone

else. He just had a feeling of having glimpsed something

very significant, but couldn't remember what it was.

Then later, when on his own, he tried to remember

what it was, and eventually succeeded in recovering it.

That's Professor Roger Penrose and his theorem that

collapsing black holes will always collapse to a singularity and

can't avoid it by kind of collapsing through themselves

in an asymmetrical way and bouncing out again, no matter

what the initial state. Accounts of his experience in books.

The key ideas may be ones that can be expressed quite simply,

which is why one can remember them. That happens in maths -

you get an idea that is in essence very simple, you can

just see it, quite intuitively. But when you try to write it

down, it may take many words to follow it all through, with

all the steps in the proof just falling out in a perfect fashion.

One wonders then how it is that one got at them, how

one knew, or at least felt so sure, that this approach

was going to work.

All htis happens in a dramatic way for some, as in Professor

Penrose's experience. But it also happens in a more plodding

kind of way. Either when trying to prove something, or

when trying to understand a proof working through line

by line.

You worry away at it, and then often at a moment of relaxation

when no longer thinking about it, one then suddenly sees

what it was about.

Or I think in even smaller ways, when going through proof

step by step, you look at a line maybe two or three times

sometimes and can't quite see how it follows from the

previuos one, then suddently do see it, and that is

kind of a bit mysterious too, how that happens.

This has often been written about by mathematicians.

However, it isn't 100% reliable. Sometimes you get a

wonderful idea in much the same fashion, but when you

try to follow it through, it just doesn't work at all.

So one can also make mistakes at this intuitive level

just as one does in the ordinary way of maths.

That has happened to me too. Doesn't seem to get quite

such a write up as the cases where it does work :-)

Robert